Dividing the questions:
So, am I right in assuming the line above is simply saying that all verbs with a Z before the ending are supposed to drop Z and the following O before adding ka or ke?
This is a rule for you to understand how an irregular-looking stem is in fact regular, just not obviously so. You would think
it was an irregular verb because the perfect doesn't look like the present, but in reality it's just a matter of applying some rules of phonology. It's a way to make it easier for us, instead of mindlessly memorizing paradigms. Short answer is Yes
Also, am I right in assuming that when a verb stem ends in a t-mute, that automatically means we are talking about the double consonant Z and only Z? Or is any verb with a t or d or th before the last letter of the verb also included?
I'm not sure I understand the question. As the rule you quoted notes, stems can end in virtually any letter (including, as it notes, double letters). There are certainly verbs whose root ends in τ and not ζ, but I suspect you were asking something else. What the example of ἀθροίζω is doing is showing that what seems like another letter (i.e., something other than τ δ θ) is in reality a τ-mute, since ζ is functioning as a double letter hiding zd/dz.
Also...almost all verbs end in o, right?
You shouldn't mix endings (to simplify, the personal endings ω εις ει ομεν ετε ουσιν) with the stem. The stem is that you get after
you've taken these away, and it's that which, with or without mutations, will be carried over to the other modes and tenses. It's what you are concerned with here.
So the stem of λέγω is what you get once you've taken the endings away: λέγω (1st person singular present indicative) without -ω is λέγ-, ὁράει [ὁρᾷ] (3rd person singular present indicative) without -ει is ὅρα- etc. In conclusion, verbs don't end in -ω (and there are even verbs that don't take -ω at all, like the -μι / athematic verbs, so called because they don't add a connecting vowel to the stem): it's just a convention to refer to them by the first person singular present indicative - and it's not even universal, where I come from we usually refer to them by the infinitive (so you would say the verb "λέγειν" instead of "λέγω", "εἶναι" instead of "εἰμι").